Volume convergence consequence of the Schoen–Yau zero mass stability conjecture

Let $\Omega_j(R)$ be the isoperimetric regions associated with a sequence satisfying the hypotheses of the zero mass stability conjecture, and let $B_0(R)\subset{\mathbb E}^3$ be the Euclidean ball of radius $R$. Volume convergence consequence. Under the hypotheses of that conjecture,

$$\operatorname{Vol}(\Omega_j(R))\to \operatorname{Vol}(B_0(R))=\frac{4\pi R^3}{3}.$$

This is presented as a consequence worth testing directly; Shi's volume inequality gives one direction, but the full convergence remains open.